Deviations from Vegard’s law in ternary III-V alloys

Vegard’s law states that, at a constant temperature, the volume of an alloy can be determined from a linear interpolation of its constituent’s volumes. Deviations from this description occur such that volumes are both greater and smaller than the linear relationship would predict. Here we use special quasirandom structures and density functional theory to investigate such deviations for $M_x{N}_{1-x}\text{As}$ ternary alloys, where $M$ and $N$ are group III species (B, Al, Ga, and In). Our simulations predict a tendency, with the exception of ${\text{Al}}_x{\text{Ga}}_{1-x}\text{As}$, for the volume of the ternary alloys to be smaller than that determined from the linear interpolation of the volumes of the $M$As and $B$As binary alloys. Importantly, we establish a simple relationship linking the relative size of the group III atoms in the alloy and the predicted magnitude of the deviation from Vegard’s law.

Figure 1

(Color online) The 32 atom $M_x{N}_{1-x}\text{As}$ SQS unit cell where $x=0.5$. This cell is extended in the $c$ direction to form a $1\times 1\times 2$ supercell containing 64 atoms. The red spheres represent As atoms, blue spheres represent $M$ atoms and green spheres represent $N$ atoms.

Figure 2

(Color online) The 64 atom $M_x{N}_{1-x}\text{As}$ SQS unit cell for $x=0.75$. The red spheres represent As atoms, blue spheres represent $M$ atoms and green spheres represent $N$ atoms.

Figure 3

(Color online) Changes in unit cell volumes as a function of $x$ in $M_x{N}_{1-x}\text{As}$ for the six random alloys. The crosses represent the unit cell volumes obtained from DFT simulations and the solid line represents a linear interpolation of the binary alloy volumes (i.e., Vegard’s law).

Figure 4

(Color online) The deviation from Vegard’s law $\left[\Delta V\left(x\right)\right]$ as a function of $x$ in $M_x{N}_{1-x}\text{As}$ for the six random ternary alloys.

Figure 5

(Color online) Examples of As lattice sites in: (a) BAs, (b) ${\text{In}}_{0.5}{\text{Ga}}_{0.5}\text{As}$, and (c) InAs. In the binary alloys, BAs and InAs, the B and In atoms form perfect tetrahedra around the As site, however, in ${\text{In}}_{0.5}{\text{Ga}}_{0.5}\text{As}$ the bond lengths are not identical and are different from the values in the constituent binary alloys.

Figure 6

(Color online) The variation in $\Delta V\left(0.5\right)$ as a function of $\left|r_M-r_N\right|$. The dotted line is the quadratic function $\Delta V\left(0.5\right)=a{\left|r_M-r_N\right|}^2+b\left|r_M-r_N\right|+c$ where $a=-26.75$, $b=-0.015$ and $c=-0.034$, with an $R^2$ value of 0.99. For comparison, the experimental data of Katayama et al. (Ref. 20) and Gehrsitz et al. (Ref. 18) have also been plotted.